Calculus: Techniques of Integration
You’ll find that there are many ways to solve an integration problem in calculus. The following list contains some handy points to remember when using different integration techniques:

Guess and Check. This technique works when the integrand is close to a simple backward derivative.

usubstitution. The integration counterpart to the chain rule; use this technique when the argument of the function you’re integrating is more than a simple x.

Integration by Parts. Integration’s counterpart to the product rule.
1. Use this technique when the integrand contains a product of functions.
2. Pick your u according to LIATE, box it, “7” it, finish it.

Trig Integrals
1. Use Pythagorean identities.
2. Use halfangle formulas.

Trigonometric Substitution. This method works when the integrand contains radicals of the forms
(or powers of these roots), where a is a constant and u is an expression in x.

Partial Fractions. This technique works for rational functions (one polynomial over another).